2020/12/08 by Benedict Leimkuhler, Leimkuhler, Benedict, Matthias Sachs +1
Computer Science · Engineering · Physics and Astronomy · #65C30 #65C40 #82M37 #Advanced Adaptive Filtering Techniques #Computation (stat.CO) #FOS: Computer and information sciences #FOS: Mathematics #FOS: Physical sciences #Gaussian Processes and Bayesian Inference #Materials Science (cond-mat.mtrl-sci) #Model Reduction and Neural Networks #Numerical Analysis (math.NA)
paper · pdf · doi:10.48550/arxiv.2012.04245
openalex publication_date 2020/12/08 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
We study the design and implementation of numerical methods to solve the generalized Langevin equation (GLE) focusing on canonical sampling properties of numerical integrators. For this purpose, we cast the GLE in an extended phase space formulation and derive a family of splitting methods which generalize existing Langevin dynamics integration methods. We show exponential convergence in law and the validity of a central limit theorem for the Markov chains obtained via these integration methods, and we show that the dynamics of a suggested integration scheme is consistent with asymptotic limits of the exact dynamics and can reproduce (in the short memory limit) a superconvergence property for the analogous splitting of underdamped Langevin dynamics. We then apply our proposed integration method to several model systems, including a Bayesian inference problem. We demonstrate in numerical experiments that our method outperforms other proposed GLE integration schemes in terms of the accuracy of sampling. Moreover, using a parameterization of the memory kernel in the GLE as proposed by Ceriotti et al [9], our experiments indicate that the obtained GLE-based sampling scheme outperforms state-of-the-art sampling schemes based on underdamped Langevin dynamics in terms of robustness and efficiency.